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W. H. Freeman and CompanyNew York

Physical Models of Living Systems

Philip NelsonUniversity of Pennsylvania

with the assistance of Sarina Bromberg,

Ann Hermundstad, and Jason Prentice

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Brief Contents

Prolog: A breakthrough on HIV 1

PART I First Steps

Chapter 1 Virus Dynamics 9

Chapter 2 Physics and Biology 27

PART II Randomness in Biology

Chapter 3 Discrete Randomness 35

Chapter 4 Some Useful Discrete Distributions 69

Chapter 5 Continuous Distributions 97

Chapter 6 Model Selection and Parameter Estimation 123

Chapter 7 Poisson Processes 153

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vi Brief Contents

PART III Control in Cells

Chapter 8 Randomness in Cellular Processes 179

Chapter 9 Negative Feedback Control 203

Chapter 10 Genetic Switches in Cells 241

Chapter 11 Cellular Oscillators 277

Epilog 299

Appendix A Global List of Symbols 303

Appendix B Units and Dimensional Analysis 309

Appendix C Numerical Values 315

Acknowledgments 317

Credits 321

Bibliography 323

Index 333

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Detailed Contents

Web Resources xvii

To the Student xix

To the Instructor xxiii

Prolog: A breakthrough on HIV 1

PART I First Steps

Chapter 1 Virus Dynamics 91.1 First Signpost 9

1.2 Modeling the Course of HIV Infection 10

1.2.1 Biological background 10

1.2.2 An appropriate graphical representation can bring

out key features of data 12

1.2.3 Physical modeling begins by identifying the key actors and

their main interactions 12

1.2.4 Mathematical analysis yields a family of predicted

behaviors 14

1.2.5 Most models must be fitted to data 15

1.2.6 Overconstraint versus overfitting 17

1.3 Just a Few Words About Modeling 17

Key Formulas 19

Track 2 21

1.2.4′ Exit from the latency period 21

1.2.6′a Informal criterion for a falsifiable prediction 21

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1.2.6′b More realistic viral dynamics models 21

1.2.6′c Eradication of HIV 22

Problems 23

Chapter 2 Physics and Biology 272.1 Signpost 27

2.2 The Intersection 28

2.3 Dimensional Analysis 29

Key Formulas 30

Problems 31

PART II Randomness in Biology

Chapter 3 Discrete Randomness 353.1 Signpost 35

3.2 Avatars of Randomness 36

3.2.1 Five iconic examples illustrate the concept of randomness 36

3.2.2 Computer simulation of a random system 40

3.2.3 Biological and biochemical examples 40

3.2.4 False patterns: Clusters in epidemiology 41

3.3 Probability Distribution of a Discrete Random System 41

3.3.1 A probability distribution describes to what extent a random

system is, and is not, predictable 41

3.3.2 A random variable has a sample space with numerical

meaning 43

3.3.3 The addition rule 44

3.3.4 The negation rule 44

3.4 Conditional Probability 45

3.4.1 Independent events and the product rule 45

3.4.1.1 Crib death and the prosecutor’s fallacy 47

3.4.1.2 The Geometric distribution describes the waiting

times for success in a series of independent trials 47

3.4.2 Joint distributions 48

3.4.3 The proper interpretation of medical tests requires an

understanding of conditional probability 50

3.4.4 The Bayes formula streamlines calculations involving

conditional probability 52

3.5 Expectations and Moments 53

3.5.1 The expectation expresses the average of a random variable

over many trials 53

3.5.2 The variance of a random variable is one measure of its

fluctuation 54

3.5.3 The standard error of the mean improves with increasing

sample size 57

Key Formulas 58

Track 2 60

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3.4.1′a Extended negation rule 60

3.4.1′b Extended product rule 60

3.4.1′c Extended independence property 60

3.4.4′ Generalized Bayes formula 60

3.5.2′a Skewness and kurtosis 60

3.5.2′b Correlation and covariance 61

3.5.2′c Limitations of the correlation coefficient 62

Problems 63

Chapter 4 Some Useful Discrete Distributions 694.1 Signpost 69

4.2 Binomial Distribution 70

4.2.1 Drawing a sample from solution can be modeled in terms of

Bernoulli trials 70

4.2.2 The sum of several Bernoulli trials follows a Binomial

distribution 71

4.2.3 Expectation and variance 72

4.2.4 How to count the number of fluorescent molecules in a

cell 72

4.2.5 Computer simulation 73

4.3 Poisson Distribution 74

4.3.1 The Binomial distribution becomes simpler in the limit of

sampling from an infinite reservoir 74

4.3.2 The sum of many Bernoulli trials, each with low probability,

follows a Poisson distribution 75

4.3.3 Computer simulation 78

4.3.4 Determination of single ion-channel conductance 78

4.3.5 The Poisson distribution behaves simply under

convolution 79

4.4 The Jackpot Distribution and Bacterial Genetics 81

4.4.1 It matters 81

4.4.2 Unreproducible experimental data may nevertheless contain an

important message 81

4.4.3 Two models for the emergence of resistance 83

4.4.4 The Luria-Delbrück hypothesis makes testable predictions for

the distribution of survivor counts 84

4.4.5 Perspective 86

Key Formulas 87

Track 2 89

4.4.2′ On resistance 89

4.4.3′ More about the Luria-Delbrück experiment 89

4.4.5′a Analytical approaches to the Luria-Delbrück

calculation 89

4.4.5′b Other genetic mechanisms 89

4.4.5′c Non-genetic mechanisms 90

4.4.5′d Direct confirmation of the Luria-Delbrück hypothesis 90

Problems 91

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Chapter 5 Continuous Distributions 975.1 Signpost 97

5.2 Probability Density Function 98

5.2.1 The definition of a probability distribution must be modified

for the case of a continuous random variable 98

5.2.2 Three key examples: Uniform, Gaussian, and Cauchy

distributions 99

5.2.3 Joint distributions of continuous random variables 101

5.2.4 Expectation and variance of the example distributions 102

5.2.5 Transformation of a probability density function 104

5.2.6 Computer simulation 106

5.3 More About the Gaussian Distribution 106

5.3.1 The Gaussian distribution arises as a limit of Binomial 106

5.3.2 The central limit theorem explains the ubiquity of Gaussian

distributions 108

5.3.3 When to use/not use a Gaussian 109

5.4 More on Long-tail Distributions 110

Key Formulas 112

Track 2 114

5.2.1′ Notation used in mathematical literature 114

5.2.4′ Interquartile range 114

5.4′a Terminology 115

5.4′b The movements of stock prices 115

Problems 118

Chapter 6 Model Selection and Parameter Estimation 1236.1 Signpost 123

6.2 Maximum Likelihood 124

6.2.1 How good is your model? 124

6.2.2 Decisions in an uncertain world 125

6.2.3 The Bayes formula gives a consistent approach to updating our

degree of belief in the light of new data 126

6.2.4 A pragmatic approach to likelihood 127

6.3 Parameter Estimation 128

6.3.1 Intuition 129

6.3.2 The maximally likely value for a model parameter can be

computed on the basis of a finite dataset 129

6.3.3 The credible interval expresses a range of parameter values

consistent with the available data 130

6.3.4 Summary 132

6.4 Biological Applications 133

6.4.1 Likelihood analysis of the Luria-Delbrück experiment 133

6.4.2 Superresolution microscopy 133

6.4.2.1 On seeing 133

6.4.2.2 Fluorescence imaging at one nanometer

accuracy 133

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6.4.2.3 Localization microscopy:

PALM/FPALM/STORM 136

6.5 An Extension of Maximum Likelihood Lets Us Infer Functional

Relationships from Data 137

Key Formulas 141

Track 2 142

6.2.1′ Cross-validation 142

6.2.4′a Binning data reduces its information content 142

6.2.4′b Odds 143

6.3.2′a The role of idealized distribution functions 143

6.3.2′b Improved estimator 144

6.3.3′a Credible interval for the expectation of

Gaussian-distributed data 144

6.3.3′b Confidence intervals in classical statistics 145

6.3.3′c Asymmetric and multivariate credible intervals 146

6.4.2.2′ More about FIONA 146

6.4.2.3′ More about superresolution 147

6.5′ What to do when data points are correlated 147

Problems 149

Chapter 7 Poisson Processes 1537.1 Signpost 153

7.2 The Kinetics of a Single-Molecule Machine 153

7.3 Random Processes 155

7.3.1 Geometric distribution revisited 156

7.3.2 A Poisson process can be defined as a continuous-time limit of

repeated Bernoulli trials 157

7.3.2.1 Continuous waiting times are Exponentially

distributed 158

7.3.2.2 Distribution of counts 160

7.3.3 Useful Properties of Poisson processes 161

7.3.3.1 Thinning property 161

7.3.3.2 Merging property 161

7.3.3.3 Significance of thinning and merging

properties 163

7.4 More Examples 164

7.4.1 Enzyme turnover at low concentration 164

7.4.2 Neurotransmitter release 164

7.5 Convolution and Multistage Processes 165

7.5.1 Myosin-V is a processive molecular motor whose stepping

times display a dual character 165

7.5.2 The randomness parameter can be used to reveal substeps in a

kinetic scheme 168

7.6 Computer Simulation 168

7.6.1 Simple Poisson process 168

7.6.2 Poisson processes with multiple event types 168

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Key Formulas 169

Track 2 171

7.2′ More about motor stepping 171

7.5.1′a More detailed models of enzyme turnovers 171

7.5.1′b More detailed models of photon arrivals 171

Problems 172

PART III Control in Cells

Chapter 8 Randomness in Cellular Processes 1798.1 Signpost 179

8.2 Random Walks and Beyond 180

8.2.1 Situations studied so far 180

8.2.1.1 Periodic stepping in random directions 180

8.2.1.2 Irregularly timed, unidirectional steps 180

8.2.2 A more realistic model of Brownian motion includes both

random step times and random step directions 180

8.3 Molecular Population Dynamics as a Markov Process 181

8.3.1 The birth-death process describes population fluctuations of a

chemical species in a cell 182

8.3.2 In the continuous, deterministic approximation, a birth-death

process approaches a steady population level 184

8.3.3 The Gillespie algorithm 185

8.3.4 The birth-death process undergoes fluctuations in its steady

state 186

8.4 Gene Expression 187

8.4.1 Exact mRNA populations can be monitored in living

cells 187

8.4.2 mRNA is produced in bursts of transcription 189

8.4.3 Perspective 193

8.4.4 Vista: Randomness in protein production 193

Key Formulas 194

Track 2 195

8.3.4′ The master equation 195

8.4′ More about gene expression 197

8.4.2′a The role of cell division 197

8.4.2′b Stochastic simulation of a transcriptional bursting

experiment 198

8.4.2′c Analytical results on the bursting process 199

Problems 200

Chapter 9 Negative Feedback Control 2039.1 Signpost 203

9.2 Mechanical Feedback and Phase Portraits 204

9.2.1 The problem of cellular homeostasis 204

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9.2.2 Negative feedback can bring a system to a stable setpoint and

hold it there 204

9.3 Wetware Available in Cells 206

9.3.1 Many cellular state variables can be regarded as

inventories 206

9.3.2 The birth-death process includes a simple form of

feedback 207

9.3.3 Cells can control enzyme activities via allosteric

modulation 207

9.3.4 Transcription factors can control a gene’s activity 208

9.3.5 Artificial control modules can be installed in more complex

organisms 211

9.4 Dynamics of Molecular Inventories 212

9.4.1 Transcription factors stick to DNA by the collective effect of

many weak interactions 212

9.4.2 The probability of binding is controlled by two rate

constants 213

9.4.3 The repressor binding curve can be summarized by its

equilibrium constant and cooperativity parameter 214

9.4.4 The gene regulation function quantifies the response of a gene

to a transcription factor 217

9.4.5 Dilution and clearance oppose gene transcription 218

9.5 Synthetic Biology 219

9.5.1 Network diagrams 219

9.5.2 Negative feedback can stabilize a molecule inventory,

mitigating cellular randomness 220

9.5.3 A quantitative comparison of regulated- and unregulated-gene

homeostasis 221

9.6 A Natural Example: The trp Operon 224

9.7 Some Systems Overshoot on Their Way to Their Stable Fixed

Point 224

9.7.1 Two-dimensional phase portraits 226

9.7.2 The chemostat 227

9.7.3 Perspective 231

Key Formulas 232

Track 2 234

9.3.1′a Contrast to electronic circuits 234

9.3.1′b Permeability 234

9.3.3′ Other control mechanisms 234

9.3.4′a More about transcription in bacteria 235

9.3.4′b More about activators 235

9.3.5′ Gene regulation in eukaryotes 235

9.4.4′a More general gene regulation functions 236

9.4.4′b Cell cycle effects 236

9.5.1′a Simplifying approximations 236

9.5.1′b The Systems Biology Graphical Notation 236

9.5.3′ Exact solution 236

9.7.1′ Taxonomy of fixed points 237

Problems 238

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Chapter 10 Genetic Switches in Cells 24110.1 Signpost 241

10.2 Bacteria Have Behavior 242

10.2.1 Cells can sense their internal state and generate switch-like

responses 242

10.2.2 Cells can sense their external environment and integrate it with

internal state information 243

10.2.3 Novick and Weiner characterized induction at the single-cell

level 243

10.2.3.1 The all-or-none hypothesis 243

10.2.3.2 Quantitative prediction for Novick-Weiner

experiment 246

10.2.3.3 Direct evidence for the all-or-none hypothesis 248

10.2.3.4 Summary 249

10.3 Positive Feedback Can Lead to Bistability 250

10.3.1 Mechanical toggle 250

10.3.2 Electrical toggles 252

10.3.2.1 Positive feedback leads to neural excitability 252

10.3.2.2 The latch circuit 252

10.3.3 A 2D phase portrait can be partitioned by a separatrix 252

10.4 A Synthetic Toggle Switch Network in E. coli 253

10.4.1 Two mutually repressing genes can create a toggle 253

10.4.2 The toggle can be reset by pushing it through a

bifurcation 256

10.4.3 Perspective 257

10.5 Natural Examples of Switches 259

10.5.1 The lac switch 259

10.5.2 The lambda switch 263

Key Formulas 264

Track 2 266

10.2.3.1′ More details about the Novick-Weiner experiments 266

10.2.3.3′a Epigenetic effects 266

10.2.3.3′b Mosaicism 266

10.4.1′a A compound operator can implement more complex

logic 266

10.4.1′b A single-gene toggle 268

10.4.2′ Adiabatic approximation 272

10.5.1′ DNA looping 273

10.5.2′ Randomness in cellular networks 273

Problems 275

Chapter 11 Cellular Oscillators 27711.1 Signpost 277

11.2 Some Single Cells Have Diurnal or Mitotic Clocks 277

11.3 Synthetic Oscillators in Cells 278

11.3.1 Negative feedback with delay can give oscillatory

behavior 278

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11.3.2 Three repressors in a ring arrangement can also oscillate 278

11.4 Mechanical Clocks and Related Devices Can also be Represented by

their Phase Portraits 279

11.4.1 Adding a toggle to a negative feedback loop can improve its

performance 279

11.4.2 Synthetic-biology realization of the relaxation oscillator 284

11.5 Natural Oscillators 285

11.5.1 Protein circuits 285

11.5.2 The mitotic clock in Xenopus laevis 286

Key Formulas 290

Track 2 291

11.4′a Attractors in phase space 291

11.4′b Deterministic chaos 291

11.4.1′a Linear stability analysis 291

11.4.1′b Noise-induced oscillation 293

11.5.2′ Analysis of Xenopus mitotic oscillator 293

Problems 296

Epilog 299

Appendix A Global List of Symbols 303A.1 Mathematical Notation 303

A.2 Graphical Notation 304

A.2.1 Phase portraits 304

A.2.2 Network diagrams 304

A.3 Named Quantities 305

Appendix B Units and Dimensional Analysis 309B.1 Base Units 310

B.2 Dimensions versus Units 310

B.3 Dimensionless Quantities 312

B.4 About Graphs 312

B.4.1 Arbitrary units 312

B.5 About Angles 313

B.6 Payoff 313

Appendix C Numerical Values 315C.1 Fundamental Constants 315

Acknowledgments 317

Credits 321

Bibliography 323

Index 333

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Web Resources

The book’s Web site (http://www.macmillanhighered.com/physicalmodels1e)

contains links to the following resources:

• The Student’s Guide contains an introduction to some computer math systems, and some

guided computer laboratory exercises.

• Datasets contains datasets that are used in the problems. In the text, these are cited like

this: Dataset 1, with numbers keyed to the list on the Web site.

• Media gives links to external media (graphics, audio, and video). In the text, these are

cited like this: Media 2, with numbers keyed to the list on the Web site.

• Finally, Errata is self-explanatory.

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